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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Haha yeah, I just added the extra stuff for any curious students and for generalisation purposes.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Here's an intuitive definition of point symmetry with diagrams: http://www.mathsisfun.com/geometry/symmetry-point.html

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent In general, a function $y=g(x)$ can have point symmetry about an arbitrary point $P(x_0,y_0)$ $\Big{(}$e.g. all cubic function graphs have point symmetry about their point of inflexion, the graph of $y=\cos^{-1}x$ has...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $(By ``point symmetry'' about the point $\left(0,\frac{\pi}{2}\right)$, we are referring to the fact that the function $f(x)=\cos^{-1}x$ satisfies $f(-x) = \pi - f(x)$ for all $x$ in the domain of $f$.)$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Here's how to visualise it. Take that region (call it $\mathcal{R}$) bounded by the curve $y=\cos ^{-1} x$, the $x$-axis, and $x=0$ and $x=-\frac{3}{4}$. Then ``reflect'' this region about the point...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Yes, that's correct. Observe that $\int \frac{x}{1+x^2} \text{ d}x = \frac{1}{2}\ln \left(1+x^2\right) + c$, for some arbitrary constant $c$ (remember to evaluate the antiderivative at the endpoints to get the value...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent The area is given by $A = \int _0 ^{\frac{\pi}{6}} \sin y\text{ d}y$.$

Re: MX2 2016 Integration Marathon This is just a standard log solution. We can simplify the sum of k's using an A.P. sum formula. Did you possibly mean product instead of sum on the denominator (makes it more interesting)?

Re: MX2 2016 Integration Marathon Yes, the answer is clearly positive as the integrand is positive throughout the domain of integration. I can't really read (or be bothered to decipher) Drsoccerball's solution though since there are too many "Paradoxica" and "Drsoccerball" references in the...

Re: HSC 2016 2U Marathon I don't know, I just thought it up. I suppose it's normal algebra. There's a lot of cool algebraic facts like this you can think up if you play around a bit, a lot of which you probably intuitively know are true actually. Here are some about averages: $\noindent...

Not necessarily. Even if a subject is "easy", it doesn't actually mean it's easy to get high marks in.

Scaling is still useful though. For really high scaling subjects, you don't need to do incredibly well to get it to scale well. For low-scaling subjects, you really do need to do very well.

Re: HSC 2016 2U Marathon Yeah

Re: HSC 2016 2U Marathon $\noindent Your idea of letting $\frac{a}{b}=\frac{c}{d}=k$ was a good one! However, we shouldn't also let $\frac{a+c}{b+d}=k$, because then we are essentially letting $\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$, which means we are assuming what we want to prove. Rather...

Re: MX2 2016 Integration Marathon $\noindent It's equivalent to the limit $\lim _{y\to \infty } \frac{\ln y}{y} = 0$, which I thought was allowed to be assumed for HSC (in which case you could use this to justify it).$

Re: HSC 2016 4U Marathon $\noindent Well done!$ $\noindent Here's the method for (i) with $a=\mathrm{e}^\lambda$.$ $\noindent Our curves are $y=\mathrm{e}^{\lambda x}$ and $y=x$. Let their point of osculation be $P\left(t,t \right)$, which clearly lies on the line $y=x$. Since it also...

Re: MX2 2016 Integration Marathon $\noindent Basically, I think you just misinterpreted the meaning of Paradoxica's notation. Those aren't log bases, they're iterated logs. Things like $\ln(\ln(\ln(\ln(x))))$.$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent The domain is (the set of all $x$ such that) $\boxed{x\leq -1\text{ or }x\geq 1}$. We have $y=\cos ^{-1}\frac{1}{x}$, so for the domain, we must have $-1\leq \frac{1}{x}\leq 1$, since the input to an inverse cosine is...

Check out this thread to see raw marks from 2015: http://community.boredofstudies.org/1177/atar-hsc-marks-class-2015/346318/submitting-raw-marks-database.html

Re: HSC 2016 4U Marathon Correct! Here's a hint that makes it easier: